There’s not really any need to do anything on the Cross, but there are a couple of options. First you could just use it to make the Sign of the Cross at the beginning and end of your repetitions of the Jesus Prayer. Another option, adapted from the Russian lestovka, is to pray, “Alleluia, alleluia, (alleluia), Glory to you, O God (x3). Glory be to the Father… Lord, have mercy (x3).” Then continue on with the Jesus Prayer. The third “alleluia” I put in parenthesis because, I believe, the Russian Old Believers only say it twice before continuing “Glory to you…”
Those are just a couple of options. There’s really nothing official, to my knowledge at least. So you’re free to add whatever devotion you like there.